The maximum value of $3 \cos \theta+4 \sin \theta$ is

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Let $f(\theta)=\sin \left(\tan ^{-1}\left(\frac{\sin \theta}{\sqrt{\cos 2 \theta}}\right)\right)$, where $-\frac{\pi}{4}<\theta<\frac{\pi}{4}$. Then the value of $\frac{d}{d(\tan \theta)}(f(\theta))$ is

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The general value of $\theta$ satisfying the equation $2 \sin ^2 \theta-3 \sin \theta-2=0$ is

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The value of $\frac{\cos 12^{\circ}-\sin 12^{\circ}}{\cos 12^{\circ}+\sin 12^{\circ}}+\frac{\sin 147^{\circ}}{\cos 147^{\circ}}$ is equal to

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If $\sin \theta+\cos \theta=m$ and $\sec \theta+\operatorname{cosec} \theta=n$, then $n(m+1)(m-1)$ is equal to

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If $x \cos \theta+y \sin \theta=\cos (\theta+\alpha)$ and $x^2+y^2=1$, then $\alpha$ is equal to

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Find the value of the following: $\tan \frac{1}{2}\left[\sin ^{-1} \frac{2 x}{1+x^2}+\cos ^{-1} \frac{1-y^2}{1+y^2}\right]$, $|x|<1

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Find the value of the following : $\tan ^{-1}(1)+\cos ^{-1}\left(-\frac{1}{2}\right)+\sin ^{-1}\left(-\frac{1}{2}\right)$.

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The range of the function $f(x)=\sin ^{-1}\left[x^2+\frac{1}{2}\right]+\cos ^{-1}\left[x^2-\frac{1}{2}\right]$, where [ ] is the greatest integer function, is:

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$\sin ^{-1}\left(\frac{x^2}{4}+\frac{y^2}{9}\right)+\cos ^{-1}\left(\frac{x}{2 \sqrt{2}}+\frac{y}{3 \sqrt{2}}-2\right)$ equals to :

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Prove that $\tan ^{-1}\left(\frac{\cos x}{1+\sin x}\right)=\frac{\pi}{4}-\frac{x}{2}, x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$

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Show that : $\sin ^{-1} \frac{12}{13}+\cos ^{-1} \frac{4}{5}+\tan ^{-1} \frac{63}{16}=\pi$.

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The maximum value of $3 \cos \theta+4 \sin \theta$ is

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